Quandle Rings Group of Theory Classical Abstract

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[5, 21, 26] for more on the historical development of the subject and its relationships with other areas of mathematics. Naturally, quandles and racks have also received a great deal of attention as purely algebraic objects. As is the case with any algebraic system, a (co)homology theory for quandles and racks has been developed in [8, 12, 13, 27], which, as applications has led to stronger invariants for knots and links. An explicit description of abelian group objects in the category of quandles and racks has been given in [18], leading to the construction of an abelian category of modules over these objects. Automorphisms of quandles have been investigated in much detail. In [14], automorphism groups of quandles of order less than 6 were determined. This investiga- tion was carried forward in [11], wherein the automorphism group of the dihedral quandle Rn was shown to be isomorphic to the group of invertible affine transformations of Zn, and the inner automorphism group of Rn was shown to be isomorphic to some dihedral group. These descriptions of automorphism groups were used to determine, up to isomorphism, all quandles of order less than 10. In [15], a description of the automorphism group of Alexander quandles was determined, and an explicit formula for the order of the automorphism group was given for finite case. In [3], some structural results are obtained for the group of automorphisms and inner automorphisms of generalised Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. This work was extended in [4], wherein several interesting subgroups of automorphism groups of conjugation quandles of groups are determined. Further, necessary and sufficient conditions are found for these subgroups to coincide with the full group of automorphisms. The purpose of the present paper is to develop a theory of quandle rings analogous to the classical theory of group rings. Attempt has been made to state the results making distinction between quandles and racks whenever possible. The main results of the paper are Theorems 3.5, 4.1, 4.3, 4.10, 5.6, 5.9 and Propositions 6.2, 6.3, 6.4, 7.2 and 7.3. Though our approach and motivation is purely algebraic, the results might be of use in knot theory. The paper is organised as follows. In Section 2, we recall some basic definitions and examples from the theory of racks and quandles. In Section 3, the main objects of our study are introduced. Given a quandle X and an associative ring R (not necessarily with unity), the quandle ring R[X] of X with coefficients in R is defined as the set of formal finite R-linear combinations of elements of X. Rack rings are defined analogously. In the first main result, Theorem 3.5, it is proved that a quandle X is 2 trivial if and only if ∆R(X)= {0}, where ∆R(X) is the augmentation ideal of R[X]. While this characterisation holds for quandles, an example is given to show that it fails for racks. In Section 4, relationships between subquandles of the given quandle and ideals of the associated quandle ring are discussed. Given a quandle X, an element x0 ∈ X and a two sided ideal I of R[X], a subquandle XI,x0 of X is defined. It is shown in Theorem 4.1 that these subquandles give a partition of the quandle X. Further, relationships among these subquandles are explored. In Theorem 4.3, it is shown that if X is a finite involutary quandle, I a two-sided ideal of R[X] ∼ and x0,y0 ∈ X two elements in the same orbit under the action of Inn(X), then XI,x0 = XI,y0 . Given a surjective quandle homomorphism f : X → Z, in Theorem 4.9, it is shown that Z is isomorphic as a quandle to a natural quotient of X. The construction of the quotient quandle leads to Theorem 4.10, which gives a correspondence between subquandles of the given quandle and ideals of the quandle ring. QUANDLE RINGS 3

Since the quandle ring R[X] does not have unity, it is desirable to embed R[X] into a ring with unity. In Section 5, the extended quandle ring R◦[X] is introduced, which is a ring with unity containing the ring R[X] as a subring. In Theorem 5.6, a short exact sequence relating certain subgroups of a group of units of R◦[X] is derived. Further, in Theorem 5.9, the structure of unit groups of R◦[X] for trivial quandles X is described. Given a quandle X and a ring R, the direct sum i i+1 XR(X) := ∆R(X)/∆R (X) i≥ X0 i i+1 of R-modules ∆R(X)/∆R (X) becomes a graded ring, called the associated graded ring of R[X]. i i+1 In Section 6, the quotients ∆R(X)/∆R (X) are investigated for dihedral quandles, and some structural results are obtained in Propositions 6.2, 6.3 and 6.4. Finally, in Section 7, we investigate weaker forms of associativity in quandle rings. A ring is called power-associative if every element of the ring generates an associative subring. In Propositions 7.2 and 7.3, we prove that quandle rings of dihedral quandles are not power- associative in general.

2. Preliminaries A rack is a non-empty set X with a binary operation (x,y) 7→ x ∗ y satisfying the following axioms: (R1) For any x,y ∈ X there exists a unique z ∈ X such that x = z ∗ y; (R2) (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) for all x,y,z ∈ X. A rack is called a quandle if the following additional axiom is satisfed: (Q1) x ∗ x = x for all x ∈ X. The axioms (R1), (R1) and (Q1) are collectively called quandle axioms. Besides knot quandles associated to knots, many interesting examples of quandles come from groups. Throughout the paper, we write arbitrary groups multiplicatively and abelian groups additively. • If G is a group, then the set G equipped with the binary operation a ∗ b = b−1ab gives a quandle structure on G, called the conjugation quandle, and denoted by Conj(G). • If A is an additive abelian group, then the set A equipped with the binary operation a ∗ b = 2b − a gives a quandle structure on A, denoted by T (A) and called the Takasaki quandle of A. For A = Z/nZ, it is called the dihedral quandle, and is denoted by Rn. • If G is a group and we take the binary operation a ∗ b = ba−1b, then we get the core quandle, denoted as Core(G). In particular, if G is additive abelian, then Core(G) is the Takasaki quandle. • Let A be an additive abelian group and t ∈ Aut(A). Then the set A equipped with the binary operation a ∗ b = ta + (idA − t)b is a quandle called the Alexander quandle of A with respect to t. Notice that, if t = −idA, then a ∗ b = 2b − a. Thus, in this case, the Alexander quandle of A is the Takasaki quandle T (A). • The preceding example can be generalised. Let G be a group and ϕ ∈ Aut(G). Then the set G equipped with the binary operation a ∗ b = ϕ(ab−1)b gives a quandle structure on G, called the generalised Alexander quandle of G with respect to ϕ. 4 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

• Let n ≥ 2 and X = {a1, a2, . . . , an} be a set with the binary operation given by ai ∗ aj = an−i+1. Then X is a rack which is not a quandle.

A quandle or rack X is called trivial if x ∗ y = x for all x,y ∈ X. Obviously, a trivial rack is a trivial quandle. Unlike groups, a trivial quandle can contain arbitrary number of elements. We denote the n-element trivial quandle by Tn. Notice that, the rack axioms are equivalent to saying that for each x ∈ X, the map Sx : X → X given by Sx(y)= y ∗ x is an automorphism of X. Further, in case of quandles, the axiom x ∗ x = x is equivalent to saying that Sx fixes x for each x ∈ X. Such an automorphism is called an inner automorphism of X, and the group generated by all such automorphisms is denoted by Inn(X). A quandle X is 2 called involutary if Sx = idX for each x ∈ X. For example, all Takasaki quandles are involutary. A loop is a set X with a binary operation (usually written multiplicatively) and an identity element such that for each a, b ∈ X, there exist unique elements x,y ∈ X such that both ax = b and ya = b. A Moufang loop is a loop X satisfying the Moufang identity (xy)(zx) = (x(yz))x for all x,y,z ∈ X. Unlike groups, Moufang loops need not be associative. In fact, an associative Moufang loop is a group. The core of a Moufang loop X is the algebraic structure with underlying set X and binary operation x ∗ y := yx−1y. Joyce [19] showed that the core of a Moufang loop is an involutary quandle. A subset Y of a rack X is called a subrack if Y is a rack with respect to the underlying binary operation. Subquandles are defined analogously. It is easy to see that a subset of an involutary quandle is a subquandle if and only if it is closed under the binary operation. The group Inn(X) acts on the quandle X in the obvious way. A quandle X is said to be connected if the action of Inn(X) on X is transitive. It is well-known that the Takasaki quandle of an abelian group A is connected if and only if A = 2A. In case of finite groups, this is equivalent to saying that A has odd order. Connected quandles are of particular importance since knot quandles are connected. Furthermore, homomorphic images of connected quandles are connected. Therefore, the quandles that appear as homomorphic images of knot quandles (under quandle colorings) are necessarily connected. Thus, classification of connected quandles is a major research theme, and has attracted a lot of attention [10, 16, 17, 25, 29].

3. Quandle rings and rack rings From this section onwards, for convenience, we denote the multiplication in a quandle or a rack by (a, b) 7→ a.b. Let X be a quandle and R an associative ring (not necessarily with unity). Let R[X] be the set of all formal ﬁnite R-linear combinations of elements of X, that is,

R[X] := αixi | αi ∈ R, xi ∈ X . i n X o QUANDLE RINGS 5

Then R[X] is an additive abelian group in the usual way. Deﬁne a multiplication in R[X] by setting

αixi . βjxj := αiβj(xi.xj). i j i,j X X X Clearly, the multiplication is distributive with respect to addition from both left and right, and R[X] forms a ring, which we call the quandle ring of X with coefficients in the ring R. Since X is non-associative, unless it is a trivial quandle, it follows that R[X] is a non-associative ring, in general. Analogously, if X is a rack, then we obtain the rack ring R[X] of X with coefficients in the ring R. We will see that, unlike groups, the quandle ring structure of trivial quandles is quite interesting. Remark 3.1. Observe that a quandle with a left multiplicative identity has only one element. For, let e ∈ X be the left identity of X. Then e.x = x for all x ∈ X. But, we have x.x = x by axiom (Q1). Now, by axiom (R1), we must have e = x for all x ∈ X, and hence X = {e}. Thus, R[X] is a non-associative ring without unity, unless X is a singleton. Analogous to group rings, we define the augmentation map ε : R[X] → R by setting

ε αixi = αi. i i X X Clearly, ε is a surjective ring homomorphism, and ∆R(X) := ker(ε) is a two-sided ideal of R[X], called the augmentation ideal of R[X]. Thus, we have

R[X]/∆R(X) =∼ R as rings. In the case R = Z, we denote the augmentation ideal simply by ∆(X). The following results is straightforward. Proposition 3.2. Let X be a rack and R an associative ring. Then {x − y | x,y ∈ X} is a generating set for ∆R(X) as an R-module. Further, if x0 ∈ X is a ﬁxed element, then the set x − x0 | x ∈ X \ {x0} is a basis for ∆R(X) as an R-module. The following is an interesting observation. Proposition 3.3. Let X be a quandle and R an associative ring. Then x.y + y.x ≡ x + y 2 mod ∆R(X) for all x,y ∈ X. Proof. By axiom (Q1), we have x2 = x in R[X] for all x ∈ X. Now, x = (x − y + y)2 = (x − y)2 + y2 + (x − y).y + y.(x − y) 2 ≡ x.y + y.x − y mod ∆R(X). 2 Thus, x.y + y.x ≡ x + y mod ∆R(X) for all x,y ∈ X. 6 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

If Y is a subrack of a rack X, then ∆R(Y ) ⊆ ∆R(X). Denote by ∆R(X : Y ) the two-sided ideal of R[X] generated by ∆R(Y ). Then

∆R(X : Y )=∆R(Y )+ R[X]∆R(Y )+∆R(Y )R[X]+ R[X]∆R(Y )R[X].

Given a subrack Y of a rack X, it is natural to look for conditions under which ∆R(Y ) is a two-sided ideal of R[X]. For trivial racks, we have the following result. Proposition 3.4. Let X be a trivial rack, Y a subrack of X and R an associative ring. Then ∆R(Y ) is a two-sided ideal of R[X].

Proof. First, notice that, ∆R(Y ) is generated as an R-module by the set {y − z | y, z ∈ Y }. Then for any i αixi ∈ R[X], we have

P αixi .(y − z)= αi(xi.y − xi.z)= αi(xi − xi) = 0 ∈ ∆R(Y ), i i i X X X and (y − z). αixi = αi(y.xi − z.xi)= αi(y − z) ∈ ∆R(Y ). i i i X X X Hence ∆R(Y ) is a two-sided ideal of R[X]. The next result characterises trivial quandles in terms of their augmentation ideals. Theorem 3.5. Let X be a quandle and R an associative ring. Then the quandle X is trivial if 2 and only if ∆R(X)= {0}. 2 Proof. Suppose that ∆R(X) = {0}. Let x0 ∈ X be an arbitrary but ﬁxed element. Observe 2 that ∆R(X) is generated as an R-module by the set (x − x0).(y − x0) | x,y ∈ X \ {x0} . It follows that (x − x ).(y − x ) = 0 for all x,y ∈ X \ {x }. In particular, (x − x ).(x − x )=0 for 0 0 0 0 0 all x ∈ X \ {x0}, which yields x − x0.x − x.x0 + x0 = 0 for all x ∈ X \{x0}. Since it is an expression in R[X], the terms must cancel oﬀ with each other. Suppose that x0.x = x and x.x0 = x0 for some x ∈ X \ {x0}. Also, we have x.x = x by axiom (Q1). Thus by axiom (R1), we must have x = x0, which is a contradiction. Hence, we must have x.x0 = x and x0.x = x0 for all x ∈ X \ {x0}. This means x0 acts trivially on all elements of X. Since x0 was an arbitrary element, it follows that the quandle X is trivial. Conversely, suppose that X is a trivial quandle. Let y,z,y′, z′ ∈ X. Then (y − z).(y′ − z′)= y.y′ − z.y′ − y.z′ + z.z′ = y − z − y + z = 0. 2 By linearity, it follows that ∆R(X)= {0}. 2 Corollary 3.6. A group G is abelian if and only if ∆R Conj(G) = {0}. 2 Remark 3.7. Obviously, if X is a trivial rack, then ∆ R(X) ={0}. However, the converse is not true for racks. For example, take X = {a1, . . . , an} with the rack structure given by ai.aj = an−i+1. Then X is not a trivial rack. On the other hand, for all 1 ≤ i, j ≤ n, we have

(ai − a1).(aj − a1)= ai.aj − ai.a1 − a1.aj + a1.a1 = an−i+1 − an−i+1 − an + an = 0, 2 and hence ∆R(X)= {0}. QUANDLE RINGS 7

4. Relations between subquandles and ideals In this section, we investigate relationships between subquandles of the given quandle and ideals of the associated quandle ring. 4.1. Subquandles associated to ideals. Let X be a quandle and R an associative ring. For each x0 ∈ X and each two sided ideal I of R[X], we deﬁne

XI,x0 = {x ∈ X | x − x0 ∈ I}.

Notice that, if I = ∆R(X), then XI,x0 = X, and if I = {0}, then XI,x0 = {x0}. In general, we have the following.

Theorem 4.1. Let X be a ﬁnite quandle and R an associative ring. Then for each x0 ∈ X and a two sided ideal I of R[X], the set XI,x0 is a subquandle of X. Further, there is a subset {x1,...,xm} of X such that X is the disjoint union

X = XI,x1 ⊔···⊔ XI,xm .

Proof. Obviously x0 ∈ XI,x0 . Further, if x,y ∈ XI,x0 , then

x.y − x0 = x.y − x0.y + x0.y − x0.x0 = (x − x0).y + x0.(y − x0) ∈ I.

This implies that x.y ∈ XI,x0 . Further, the inner automorphism Sy restricts to a map XI,x0 →

XI,x0 . Since Sy is injective and X is ﬁnite, it follows that Sy XI,x0 = XI,x0 . Hence, there exists a unique element z ∈ XI,x0 such that x = z.y, thereby proving that XI,x0 is a subquandle of X. For the second assertion, it is suﬃcient to prove that if x0,y0 ∈ X, then the subquandles XI,x0 and XI,y0 intersect if and only if they are equal. Let z ∈ XI,x0 ∩XI,y0 . Then we have x0 −y0 ∈ I, which further implies that x0 ∈ XI,y0 . Now, if x ∈ XI,x0 , then x − y0 = x − x0 + x0 − y0 ∈ I, which implies that XI,x0 ⊆ XI,y0 . By interchanging roles of x0 and y0, we get XI,x0 = XI,y0 .

Remark 4.2. If X is an involutary quandle (not necessarily ﬁnite), then XI,x0 is always a subquandle of X, being closed under the quandle multiplication. In particular, this holds for trivial quandles.

Given a two sided ideal I of R[X] and elements x0,y0 ∈ X, it is natural to ask whether there is any relation between the subquandles XI,x0 and XI,y0 . We answer this question in the following result. Theorem 4.3. Let X be a ﬁnite involutary quandle and R an associative ring. If I is a two-sided ∼ ideal of R[X] and x0,y0 ∈ X are in the same orbit under action of Inn(X), then XI,x0 = XI,y0 .

Proof. We ﬁrst claim that if x0,y0 ∈ X, then the map fy0 : XI,x0 → XI,x0.y0 given by fy0 (x)= x.y0 is an injective quandle homomorphism. For, if x ∈ XI,x0 , then x − x0 ∈ I. Consequently, x.y0 − x0.y0 = (x − x0).y0 ∈ I, which further implies fy0 (x) = x.y0 ∈ XI,x0.y0 . The claim now follows by observing that fy0 is simply the restriction of the inner automorphism Sy0 on the subquandle XI,x0 . Now, suppose that x0,y0 ∈ X such that there exists f ∈ Inn(X) with f(x0)= y0. Since X is −1 involutary, Sx = Sx for all x ∈ X. Therefore, by deﬁnition f = Sxk · · · Sx1 for some xi ∈ X, and

y0 = Sxk · · · Sx1 (x0) = (· · · ((x0.x1).x2) · · · ).xk. 8 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

By the claim above, there is a sequence of embeddings of subquandles of X

. XI,x0 ֒→ XI,x0.x1 ֒→ XI,(x0.x1).x2 ֒→··· ֒→ XI,y0

Thus, we obtain an embedding of XI,x0 into XI,y0 . Writing x0 = Sx1 · · · Sxk (y0), we obtain an ∼ embedding of XI,y0 into XI,x0 . Since X is ﬁnite, it follows that XI,x0 = XI,y0 . The following is an immediate consequence. Corollary 4.4. Let X be a ﬁnite connected involutary quandle and R an associative ring. Then ∼ XI,x0 = XI,y0 for any ideal I of R[X] and x0,y0 ∈ X.

For example, trivial quandles and dihedral quandles Rn of odd order are connected and ∼ involutary. The example below shows that we cannot write XI,x0 = XI,y0 instead XI,x0 = XI,y0 in Theorem 4.3.

Example 4.5. Let T2 = {x,y} be the two element trivial quandle and I the two-sided ideal in Z[T2], generated by the element u = 2x + 2y. Since ux = u, uy = u, xu = 4x, yu = 4y, it is not diﬃcult to see that I = 2αx + (2α + 4β)y | α, β ∈ Z .

Obviously x ∈ (T2)I,x. Now, if y ∈(T2)I,x, then y −x ∈ I, i.e. y − x = 2αx+(2α+4β)y for some integers α and β, which is not possible. This implies (T2)I,x = {x}. Similarly (T2)I,y = {y}, and therefore (T2)I,x 6= (T2)I,y.

Notice that, in the preceding example, we have I = 2Z · I1, where 2Z is an ideal of Z and

I1 = {αx + (α + 2β)y | α, β ∈ Z} is a proper ideal of Z[T2] since y 6∈ I1. Let X be a quandle and R an associative ring. We say that an ideal I of the rack ring R[X] is R-prime if I is not a product I = I0 · I1, where I0 is a proper ideal of R and I1 is a proper ideal of R[X]. In view of this deﬁnition, it is tempting to ask whether the condition of I being

R-prime in Theorem 4.3 imply that XI,x0 = XI,y0 ? The following example shows that it is not true in general.

Example 4.6. Let X = R4 = {a0, a1, a2, a3} be the dihedral quandle. We know that X is not connected since the elements a0 and a1 lying in diﬀerent orbits. In the quandle ring Z[R4] take 2 the two-sided ideal I with the linear basis {a0 + a1 − a2 − a3, 2(a2 − a0)} (in fact I =∆ (R4)). It is not diﬃcult to check that XI,x0 = {x0} and XI,y0 = {y0}, i. e. XI,x0 6= XI,y0 . Moreover, X is the following disjoint union of its trivial subquandles:

X = XI,a0 ⊔ XI,a1 ⊔ XI,a2 ⊔ XI,a3 . Remark 4.7. It is worth noting that the results of the preceding discussion holds for racks as well. It would be interesting to explore whether Theorem 4.3 holds if X is not involutary or if X is involutary but x0 and y0 lie in diﬀerent orbits under the action of Inn(X). QUANDLE RINGS 9

4.2. Ideals associated to subquandles. Next, we proceed in the reverse direction of associ- ating an ideal of R[X] to a subquandle of X. Let f : X → Z be a quandle homomorphism. Consider the equivalence relation ∼ on X given by x1 ∼ x2 if f(x1) = f(x2). Let X/∼ be the set of equivalence classes, where equivalence class of an element x is denoted by ′ ′ Xx := {x ∈ X | f(x )= f(x)}.

Proposition 4.8. Xx is a subquandle of X for each x ∈ X.

Proof. Obviously Xx is non-empty since x ∈ Xx. Only the axiom (R1) needs to be checked. Let x1,x2 ∈ Xx. Then f(x1.x2)= f(x1).f(x2)= f(x).f(x)= f(x), and hence x1.x2 ∈ Xx. Further, if x3 ∈ X is the unique element such that x3.x1 = x2, then applying f yields f(x3).f(x)= f(x). This together with the axiom (Q1) imply that f(x3)= f(x), and hence x3 ∈ Xx. The following is a sort of ﬁrst isomorphism theorem for quandles.

Theorem 4.9. The binary operation given by Xx1 ◦ Xx2 = Xx1.x2 gives a quandle structure on X/∼. Further, if f : X → Z is a surjective quandle homomorphism, then X/∼ =∼ Z as quandles.

Proof. The operation Xx1 ◦ Xx2 = Xx1.x2 is clearly well-defined. We only need to check the axiom (R1). Let Xx1 , Xx2 ∈ X/∼. If x3 ∈ X is the unique element such that x3.x1 = x2, then f(x3).f(x1) = f(x2) and Xx3 ◦ Xx1 = Xx2 . Suppose that there exists another element ′ ′ ′ Xx3 ∈ X/∼ such that Xx3 ◦ Xx1 = Xx2 , then f(x3).f(x1)= f(x2). By the axiom (Q1), we must ′ ′ have f(x3)= f(x3), and hence Xx3 = Xx3 . Suppose that f : X → Z is surjective. By definition of the equivalence relation, there is a ¯ ¯ well-defined bijective map f : X/∼ → Z given by f(Xx)= f(x). If Xx1 , Xx2 ∈ X/∼, then ¯ ¯ ¯ ¯ f(Xx1 ◦ Xx2 )= f(Xx1.x2 )= f(x1.x2)= f(x1).f(x2)= f(Xx1 ).f(Xx2 ), and hence f¯ is an isomorphism of quandles. We know that a subgroup of a group is normal if and only if it is the kernel of some group homomorphism. In a similar way, we say that a subquandle Y of a quandle X is normal if

Y = Xx0 for some x0 ∈ X and some quandle homomorphism f : X → Z. In this case, we say that Y is normal based at x0. A pointed quandle, denoted (X,x0), is a quandle X together with a ﬁxed base point x0. Let f : (X,x0) → (Z, z0) be a homomorphism of pointed quandles, and Y = Xx0 a normal subquandle based at x0. In this situation, we consider Y as the base point of X/∼, and denote X/∼ by X/Y . Then the natural map x 7→ Xx is a surjective homomorphism of pointed quandles

(X,x0) → (X/Y, Xx0 ). This further extends to a surjective ring homomorphism, say, π : R[X] → R[X/Y ] with ker(π) being a two sided ideal of R[X]. Let (X,x0) be a pointed quandle, I the set of two sided ideals of R[X] and S the set of normal subquandles of X based x0. Then there exist maps Φ : I → S given by

Φ(I)= XI,x0 10 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH and Ψ : S→I given by Ψ(Y ) = ker(π). With this set up, we have the following.

Theorem 4.10. Let (X,x0) be a pointed quandle and R an associative ring. Then ΦΨ = idS and ΨΦ 6= idI.

Proof. Let Y = Xx0 be a normal subquandle of X, that is, Y = {x ∈ X | f(x)= f(x0)}. Then ΦΨ(Y ) = Φ ker(π) = x ∈ X | x − x0 ∈ ker(π) = x ∈ X | Xx = Xx0 = x ∈ X | f(x)= f(x 0) = Y. Obviously, ΨΦ 6= idI , since ΨΦ R[X] = Ψ(X)=∆R(X) 6= R[X]. 5. Extended rack ring and units In this section, we assume that R is an associative ring with unity 1. Let X be a rack. Since R[X] is a ring without unity, it is desirable to embed R[X] into a ring with unity. The ring R◦[X]= R[X] ⊕ Re, where e is a symbol satisfying e i αixi = i αixi = i αixi e, is called the extended rack ring of X. For convenience, we denote the unity 1e of R◦[X] by e. We extend the augmentation map ε : R◦[X] → R to obtain the P extended augmentationP P ideal ◦ ∆R◦ (X) := ker(ε : R [X] → R).

In the case R = Z, we simply denote it by ∆◦(X). As before, it is easy to see that the set {x − e | x ∈ X} is a basis for ∆R◦ (X) as an R-module.

Proposition 5.1. If X is a rack and x0 ∈ X a ﬁxed element, then ∆R◦ (X)=∆R(X)+R(e−x0).

◦ Proof. Let u ∈ ∆R (X). Then u = i αixi + βe, where i αi + β = 0 and αi, β ∈ R. Thus, we can rewrite u = αi(xi − x0)+ β(e − x0), where αi(xi − x0) ∈ ∆R(X). Thus every element i P i P u ∈ ∆R◦ (X) can be written as u = a + β(e − x0), where a ∈ ∆R(X) and β ∈ R. P P 2 ◦ Proposition 5.2. If X is a quandle, then ∆R◦ (X)=∆R (X).

2 ◦ Proof. Obviously we have ∆R◦ (X) ⊆ ∆R (X). To prove the opposite inclusion, notice that, 2 ∆R◦ (X) is generated by {(xi − e).(xj − e) | xi,xj ∈ X}. Taking xj = xi, we have

(xi − e).(xi − e)= −(xi − e) ∈ ∆R◦ (X).

◦ 2 Hence ∆R (X) ⊆ ∆R◦ (X), and proposition is proved.

2 ◦ Remark 5.3. If X is a rack with ∆R◦ (X)=∆R (X), then it is not necessarily a quandle. For example, if X = {a, b} with the rack structure a.a = a.b = b and b.a = b.b = a, then 2 ◦ ∆R◦ (X)=∆R (X), but X is not a quandle. QUANDLE RINGS 11

5.1. Units in extended rack rings. Let X be a rack and R an associative ring with unity 1. Though the ring R◦[X] has unity, it is non-associative, in general. Therefore, a natural problem is to determine maximal multiplicative subgroups of R◦[X]. Let U(R◦[X]) denote a maximal multiplicative subgroup of the ring R◦[X]. Notice that, ε : R◦[X] → R maps U(R◦[X]) onto R∗, the group of units of R. Let ◦ ◦ U1(R [X]) := r ∈U(R [X]) | ε(r) = 1 , ◦ ∗ ◦ be the subgroup of normalized units. Then U(R [X]) = R U1(R [X]), and one only need to compute the group of normalized units. Define V(R◦[X]) := e + a ∈U(R◦[X]) | a ∈ R[X] . ◦ ◦ ◦ Then it is not difficult to see that V(R[X]) is a normal subgroup of U(R [X]) and U(R [X]) = R∗ V(R◦[X]). To understand V(R◦[X]) further, we define ◦ ◦ ◦ V1(R [X]) := U1(R [X]) ∩V(R [X]). ◦ Proposition 5.4. Let X be a rack and R an associative ring with unity. Then V1(R [X]) = ◦ ◦ e + a ∈U(R [X]) | a ∈ ∆R(X) and is a normal subgroup of U(R [X]). ◦ ◦ Proof. Let r = e + a ∈ V(R [X]). Then ε(r)=1+ ε(a). On the other side, if r ∈ U1(R [X]), then ε(r) = 1. Hence ε(a) = 0, i. e. a ∈ ∆(R[X]), and the first assertion is proved. ◦ ◦ −1 −1 −1 Let u ∈U(R [X]) and r = e + a ∈V1(R [X]). Then u .r.u = e + u .a.u and ε(u .r.u)= −1 −1 −1 ◦ 1+ ε(u .a.u), where ε(u .a.u) = ε(a) = 0. Hence u .r.u ∈ V1(R [X]) proving the second assertion.

Let X be a rack and x0 ∈ X a ﬁxed element. Deﬁne the set ◦ ∗ V2(R [X]) = e + (λ − 1)x0 | λ ∈ R }.

Proposition 5.5. Let X be a rack, x0 ∈ X a ﬁxed element and R an associative ring with ◦ ◦ ∗ unity. Then V2(R [X]) is a subgroup of V(R [X]) and is isomorphic to R . ◦ −1 −1 ◦ Proof. Notice that, if r = e+(λ−1)x0 ∈V2(R [X]), then r = e+(λ −1)x0 ∈V2(R [X]), and ◦ hence r ∈V(R [X]). Further, if r = e+(λ−1)x0 and s = e+(µ−1)x0, then rs = e+(λµ−1)x0. ◦ ◦ Hence V2(R [X]) is a subgroup of V(R [X]). Finally, the map λ 7→ e + (λ − 1)x0 gives an ∗ ◦ isomorphism of R onto V2(R [X]).

Theorem 5.6. Let X be a rack, x0 ∈ X a ﬁxed element and R an associative ring with unity. Then there exists a split exact sequence ◦ ◦ ◦ 1 −→ V1(R [X]) −→ V(R [X]) −→ V2(R [X]) −→ 1. Proof. Let v = e + a ∈ V(R◦[X]). Then ε(v)=1+ ε(a) ∈ R∗. Since a ∈ R[X], we have n n a = i=0 αixi for αi ∈ R, xi ∈ X and ε(a)= i=0 αi. Rewrite a in the form n n n P P a = αi(xi − x0)+ αi x0 = αi(xi − x0) + (ε(v) − 1)x0, i i ! i X=1 X=0 X=1 where n ∗ αi(xi − x0) ∈ ∆R(X) and ε(v) ∈ R . i X=1 12 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

Hence, we can write

v = e + a0 + (ε(v) − 1)x0, a0 ∈ ∆R(X). Now deﬁne a map ◦ ◦ ϕ : V(R [X]) −→ V2(R [X]) by the rule ϕ(e + a0 + (ε(v) − 1)x0) = e + (ε(v) − 1)x0. If v = e + a0 + (ε(v) − 1)x0 and ◦ u = e + b0 + (ε(u) − 1)x0 are elements of V(R [X]), then

v.u = e + a0 + b0 + (ε(v) − 1)x0.b0 + (ε(u) − 1)a0.x0 + a0.b0 + (ε(v)ε(u) − 1)x0 ◦ and ϕ(v.u)= e + (ε(v)ε(u) − 1)x0 = ϕ(v).ϕ(u). Clearly, ϕ is surjective and ker(ϕ)= V1(R [X]). Thus we obtain the short exact sequence ◦ ◦ ◦ 1 −→ V1(R [X]) −→ V(R [X]) −→ V2(R [X]) −→ 1, which splits by Proposition 5.5.

As a consequence, we have the following.

Corollary 5.7. Let X be a rack, x0 ∈ X and R an associative ring with unity. Then an ◦ arbitrary element u = e + a0 + (λ − 1)x0 ∈ V(R [X]), λ ∈ R∗, is the product u = u1u2, where −1 ◦ ◦ u1 = e + a0 e + (λ − 1)x0 ∈V1(R [X]) and u2 = e + (λ − 1)x0 ∈V2(R [X]). ◦ ∗ ◦ Since V2(R [X]) =∼ R , it follows from Theorem 5.6 that if we know the structure of V1(R [X], then we can determine the structure of V(R◦[X]. The preceding discussion can be summarised in the following digram.

U(R◦[X]) ♣7 j❱❱❱ ♣♣ ❱❱❱❱ ♣♣♣ ❱❱❱❱ ♣♣ ❱❱❱❱ ♣♣♣ ❱❱❱ ◦ ◦ ◦ ◦ U1(R [X]) V(R [X]) = V1(R [X]) ⊕V2(R [X]) g◆◆ ❤❤4 ◆◆ ❤❤❤❤ ◆◆◆ ❤❤❤❤ ◆◆◆ ❤❤❤❤ ◆ ❤❤❤❤ ◦ V1(R [X])

5.2. Units in extended rack rings of trivial racks. Notice that, if T is a trivial rack and R an associative ring with unity, then the ring R◦[T] is associative with unity, and hence U(R◦[T]) is precisely the group of units of R◦[T]. As observed in the preceding section, the main problem ◦ ◦ in determining U(R [T ]) is the description of V1(R [T ]). Proposition 5.8. Let T be a trivial rack and R an associative ring with unity. Then ◦ V1(R [T]) = e + a | a ∈ ∆R(T) ◦ ◦ is an abelian subgroup of U1(R [T]). Further, V1(R [T]) =∼ ∆R(T) . QUANDLE RINGS 13

Proof. Let a, b ∈ ∆R(T). Then (e + a).(e + b) = e + a + b + a.b = e + a + b = (e + b).(e + a), 2 ◦ since ∆R(T) = {0} by Theorem 3.5. This implies that V1(R [T]) is closed under multiplication ◦ and is abelian. Further, (e + a).(e − a)= e + a − a +0= e shows that V1(R [T]) is an abelian ◦ ◦ subgroup of U1(R [X]). Finally, the map e + a 7→ a gives the desired isomorphism of V1(R [T]) onto the additive group ∆R(T). More generally, we prove the following.

Theorem 5.9. Let T be a trivial rack, x0 ∈ T and R an associative ring with unity. Then ◦ ∗ U1(R [T]) = e + a + α(x0 − e) | a ∈ ∆R(T) and α − 1 ∈ R . ◦ ◦ Proof. Let u∈ U1(R [T]). Then ε(u) = 1, and hence −e + u ∈ ∆R(T). This implies that ◦ u = e + w for some w = i αi(xi − e) ∈ ∆R(T). Now let x0 ∈ T be a ﬁxed element. Then we can write P u = e + αi(xi − e)= e + αi(xi − x0)+ αi(x0 − e), i i X Xi=06 X where i=06 αi(xi −x0) ∈ ∆R(T). Thus we have written u = e+a+α(x0 −e) for some a ∈ ∆R(T) and α ∈ R. Since u is a unit, there exists an elementu ¯ = e + b + β(x − e) such that u.u¯ = 1. P 0 Then equating the two sides gives

a + (1 − α)b + (α + β − αβ)(x0 − e) = 0, which in turn gives a = (α − 1)b and α + β = αβ. Clearly, α 6= 1, and hence the only solution α ∗ is β = α−1 , which is possible iﬀ that α − 1 ∈ R . ∗ Conversely, consider an element u = e + a + α(x0 − e), where a ∈ ∆R(T) and α − 1 ∈ R . 1 α Takingu ¯ = e + α−1 a + α−1 (x0 − e), we easily see that u.u¯ =u.u ¯ = 1. Henceu ¯ is the inverse of u, and the proof is complete. As a consequence, we have the following for integral coeﬃcients.

Corollary 5.10. Let T be a trivial rack, x0 ∈ X and R an associative ring with unity. Then the following statements hold: ◦ (1) U(Z [T]) = ± e + a + α(x0 − e) | a ∈ ∆R(T) and α = 0, 2 . (2) If T = {x }, then U(Z◦[T ]) = {±e, 2x − e, −2x + e} =∼ Z ⊕ Z . 1 0 1 2 2 ◦ ◦ Next we consider nilpotency of V(R [T]). Obviously, V(R [T1]) is nilpotent. In the general case, we prove the following. Proposition 5.11. Let R be an associative ring with unity such that |R∗| > 1. If T is a trivial rack with more than one element, then V(R◦[T]) is not nilpotent. ◦ Proof. It is enough to prove that for the two element trivial rack T2 = {x,y} the group V(R [T2]) is not nilpotent. Let v = e + (y − x) + (ε(v) − 1)x and u = e + (y − x) + (ε(u) − 1)x ◦ be two elements of V(R [T2]) with ε(v) 6= ε(u) and ε(u) 6= 1. Then v−1 = e − ε(v)−1(y − x) + (ε(v)−1 − 1)x, u−1 = e − ε(u)−1(y − x) + (ε(u)−1 − 1)x. 14 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

Now consider the following sequence of elements

w1 = [v, u] and wn = [wn−1, u] for n> 1. We have

w1 = [v, u]= e + (ε(u) − ε(v))(y − x). Using induction on n, we get n−1 wn = e + (ε(u) − ε(v))(ε(u) − 1) (y − x). ◦ Since ε(v) 6= ε(u) and ε(u) 6= 1, the elements wn are not trivial, and hence V(R [T]) is not nilpotent.

An important ingredient in the study of units of group rings are the central units. We deﬁne the center Z(R[X]) of the rack ring R[X] as Z(R[X]) := u ∈ R[X] | u.v = v.u for all v ∈ R[X] . Notice that, Z(R[X]) is clearly an additive subgroup of the rack ring R[X]. Since R[X], in general, is non-associative, it follows that Z(R[X]) need not be a subring of R[X]. Of course, if X is a trivial rack, then Z(R[X]) is a subring of R[X]. However, even in this case, Z(R[X]) need not be an ideal of R[X]. Recall that, a rack is called latin if the left multiplication by each element is a permutation of the rack. For example, odd order dihedral quandles are latin. Notice that, a latin rack is always connected.

Proposition 5.12. Let X = {x1,x2,...,xn} be a ﬁnite latin rack. Then the following hold:

(1) The element w = x1 + x2 + · · · + xn lies in the center Z(R[X]). 1 (2) If R is a ﬁeld with characteristic not dividing n, then n w is idempotent. (3) If R is a ﬁeld with characteristic not dividing n, then any element of the form e + αw α with α 6= −1/n is a central unit with inverse e − 1+nα w.

Proof. Notice that, for each xi, we have

w.xi = Sxi (x1)+ Sxi (x2)+ · · · + Sxi (xn)= w, since Sxi is an automorphism of X and acts as a permutation. On the other hand, since the rack X is latin, we have xi.w = w proving (1). If the characteristic of R does not divide n, then a direct computation yields (2). For (3), suppose that e + αw has inverse e + βw. Then, using (2), we have (e + αw).(e + βw)= e + (α + β + nαβ)w = e. Consequently, α+β+nαβ = 0, and hence β = −α/(1+nα). By (1), e+αw is clearly central. QUANDLE RINGS 15

6. Associated graded ring of a rack ring Let X be a rack and R an associative ring. Consider the direct sum i i+1 XR(X) := ∆R(X)/∆R (X) i≥ X0 i i+1 of R-modules ∆R(X)/∆R (X). We regard XR(X) as a graded R-module with the convention i i+1 that the elements of ∆R(X)/∆R (X) are homogeneous of degree i. Define multiplication in i i+1 i i+1 XR(X) as follows. For ui ∈ ∆R(X), let ui = ui +∆R (X). Then, for ui ∈ ∆R(X)/∆R (X), j j+1 uj ∈ ∆R(X)/∆R (X), we define ui ·uj = uiuj. The product of two arbitrary elements of XR(X) is defined by extending the above product by linearity, that is, if u = ui and v = vj are decompositions of u, v ∈XR(X) into homogeneous components, then P P u · v = ui · vj. i,j X With this multiplication, XR(X) becomes a graded ring, and we call it the associated graded ring of R[X]. For trivial quandles we have a full description of its associated graded ring.

Proposition 6.1. If T is a trivial quandle and x0 ∈ T , then XR(T) = Rx0 ⊕ ∆R(T).

Proof. It follows from the deﬁnition of XR(T) and Theorem 3.5. i i+1 For studying XR(X) we need to understand the quotients ∆R(X)/∆R (X). In the case of groups, we have the following result (see [28, p. 122]): Let G be a ﬁnite group and Qn(G) = n n+1 ∆Z(G)/∆Z (G), then there exist integers n0 and π such that ∼ Qn(G) = Qn+π(G) for all n ≥ n0. In what follows, we compute powers of the integral augmentation ideals of the dihedral quandles Rn for some small values of n. Observe that R2 =T2, the trivial quandle with 2 elements. Consider the integral quandle ring of the dihedral quandle R3 = {a0, a1, a2}. Set e1 := a1 − a0 2 and e2 := a2 − a0. Then ∆(R3)= he1, e2i. To determine ∆ (R3), we compute the products eiej and put them in the following table.

· e1 e2 e1 e1 − 2e2 −e1 − e2 e2 −e1 − e2 −2e1 + e2 2 2 It follows from the table that ∆ (R3) = he1 + e2, 3e2i and ∆(R3)/∆ (R3) =∼ Z3. Using induction we prove the following result. Proposition 6.2. The following holds for each natural number k: 2k−1 k−1 k−1 2k k−1 k k k+1 ∆ (R3)= h3 e1, 3 e2i, ∆ (R3)= h3 (e1 + e2), 3 e2i, and ∆ (R3)/∆ (R3) =∼ Z3. From the preceding proposition, we obtain the inﬁnite ﬁltration 2 3 ∆(R3) ⊇ ∆ (R3) ⊇ ∆ (R3) ⊇ . . . n n such that ∩n≥0∆ (R3)= {0}. Hence, ∆ (R3) in not nilpotent, but is residually nilpotent. 16 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

Next, we calculate the powers of augmentation ideal of R4. First, notice that

Z[R4]= Za0 ⊕ Za1 ⊕ Za2 ⊕ Za3 and ∆(R4)= Z(a1 − a0) ⊕ Z(a2 − a0) ⊕ Z(a3 − a0). Let us set e1 := a1 − a0, e2 := a2 − a0, e3 := a3 − a0. 2 Proposition 6.3. (1) ∆ (R4) is generated as an abelian group by the set {e1 − e2 − e3, 2e2} 2 and ∆(R4)/∆ (R4) =∼ Z ⊕ Z2. k k−1 k (2) If k> 2, then ∆ (R4) is generated as an abelian group by the set {2 (e1 −e2 −e3), 2 e2} k−1 k and ∆ (R4)/∆ (R4) =∼ Z2 ⊕ Z2. 2 Proof. First we consider (1). The abelian group ∆ (R4) is generated by the products ei.ej. We write these product in the following table.

· e1 e2 e3 e1 e1 − e2 − e3 0 e1 − e2 − e3 e2 −2e2 0 −2e2 e3 −e1 − e2 + e3 0 −e1 − e2 + e3

2 Hence, ∆ (R4) is generated by the set

{e1 − e2 − e3, −2e2, −e1 − e2 + e3} 2 2 3 and {e1 − e2 − e3, 2e2} form a basis of ∆ (R4). The fact that ∆ (R4)/∆ (R4) =∼ Z2 ⊕ Z2 follows from the properties of abelian groups. 3 2 Next, we consider (2). To ﬁnd a basis of ∆ (R4), multiply the basis of ∆ (R4) by the elements e1, e2, e3. Using the multiplication table, we get

(e1 − e2 − e3).e1 = 2(e1 + e2 − e3), e1.(e1 − e2 − e3) = 0, (e1 − e2 − e3).e2 = e2.(e1 − e2 − e3) = 0,

(e1 − e2 − e3).e3 = 2(e1 + e2 − e3), e3.(e1 − e2 − e3) = 0, 2e1.e2 = 0, 2e2.e1 = −4e2,

2e2.e2 = 0, 2e2.e3 = −4e2, 2e3.e2 = 0. 3 Hence, the set {2(e1 − e2 − e3), 4e2} forms a basis of ∆ (R4). We obtain the general formulas using induction on k.

Finally, we consider the quandle R5. Recall that

Z[R5]= Za0 ⊕ Za1 ⊕ Za2 ⊕ Za3 ⊕ Za4, and ∆(R4)= Z(a1 − a0) ⊕ Z(a2 − a0) ⊕ Z(a3 − a0) ⊕ Z(a4 − a0). Denote ei := ai − a0 for i = 1, 2, 3, 4. 2 Proposition 6.4. ∆ (R5) is generated as an abelian group by {e1 −e2 −e4, e2 +2e4, e3 +3e4, 5e4} 2 and ∆(R5)/∆ (R5) =∼ Z5. QUANDLE RINGS 17

2 Proof. The abelian group ∆ (R5) is generated by the products ei.ej. We write these products in the following table.

· e1 e2 e3 e4 e1 e1 − e2 − e4 e3 − 2e4 −e1 − e4 e2 − e3 − e4 e2 −e2 − e3 e2 − e3 − e4 −e1 − e3 + e4 e1 − 2e3 e3 −2e2 + e4 e1 − e2 − e4 −e1 − e2 + e3 −e2 − e3 e4 −e1 − e2 + e3 −e1 − e4 −2e1 + e2 −e1 − e3 + e4

2 It follows that ∆ (R5) is generated by the elements from the above table. Further, it is not diﬃcult to see that the elements

{e1 − e2 − e4, e2 + 2e4, e3 + 3e4, 5e4} 2 2 form a basis of ∆ (R5). The fact that ∆(R5)/∆ (R5) =∼ Z5 follows from the properties of abelian groups. The preceding results lead us to formulate the following.

Conjecture 6.5. Let Rn be the dihedral quandle and R a ring. k k+1 (1) If n> 1 is an odd integer, then ∆ (Rn)/∆ (Rn) =∼ Zn for all k ≥ 1. k k+1 (2) If n> 2 is an even integer, then |∆ (Rn)/∆ (Rn)| = n for all k ≥ 2.

7. Power-associativity of quandle rings We know that any trivial quandle is associative, but an arbitrary quandle, and hence its quandle ring need not be associative. In particular, the dihedral quandle R3, and hence all dihedral quandles Rn, n ≥ 3 are not associative. Also, it is easy to see that the conjugation quandle Conj(G) is associative if and only if G is abelian, i.e. Conj(G) is a trivial quandle. For core quandles, we prove the following Proposition 7.1. A core quandle Core(G) is associative if and only if G has exponent 2. Proof. Let a, b, c ∈ G. Then, in Core(G), we have (a.b).c = cb−1ab−1c and a.(b.c)= cb−1ca−1cb−1c. Now, it follows that Core(G) is associative if and only if (ca−1)2 = 1 i.e. G has exponent 2. Recall that, a ring R is called power-associative if every element of R generates an associative subring of R (see [1]). If T is a trivial quandle and R an associative ring, then R[T] is associative, and hence power-associative. In general, it is an interesting question to determine the conditions under which the ring R[X] is power-associative. We investigate power-associativity of quandle rings of dihedral quandles Rn. The cases n = 1, 2 are obvious. For n = 3, we prove the following result. Proposition 7.2. Let R be a commutative ring with unity of characteristic not equal to 2, 3 or 5. Then the quandle ring R[R3] is not power-associative. 18 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

2 2 Proof. Let R3 = {a0, a1, a2}. Notice that, R[R3] is a commutative ring, and hence u .u = u.u 2 2 2 for all u ∈ R[R3]. On the other hand, we show that the equality (u .u).u = u .u does not hold for all u ∈ R[R3]. Let u = αa0 + βa1 + γa2, where α, β, γ ∈ R. We compute 2 2 2 2 u = (α + 2βγ)a0 + (β + 2αγ)a1 + (γ + 2αβ)a2. and 2 2 4 2 2 3 3 2 4 2 2 3 3 2 u .u = (α + 6β γ + 4αβ + 4αγ + 12α βγ)a0 + (β + 6α γ + 4α β + 4βγ + 12αβ γ)a1+ 4 2 2 3 3 2 +(γ + 6α β + 4α γ + 4β γ + 12αβγ )a2. On the other hand 2 3 2 2 2 2 3 2 2 2 2 u .u = (α + 2αβγ + βγ + 2αβ + γβ + 2αγ )a0 + (β + 2α β + αγ + 2αβγ + α γ + 2βγ )a1+ 3 2 2 2 2 +(γ + 2αβγ + αβ + 2α γ + α β + 2β γ)a2, and 2 4 2 2 2 2 2 2 2 3 3 3 3 (u .u).u = (α + 3α β + 3α γ + 3αβγ + 3αβ γ + 6α βγ + 3βγ + αβ + 3β γ + αγ )a0+ 4 2 2 2 2 2 2 2 3 3 3 3 +(β + 3αβγ + 3α βγ + 6αβ γ + 3α β + 3β γ + 3αγ + α β + 3α γ + βγ )a1+ 4 2 2 2 2 2 2 2 3 3 3 3 +(γ + 3α γ + 3β γ + 3αβ γ + 3α βγ + 6αβγ + 3αβ + α γ + 3α β + β γ)a2. It follows that R[R3] is not power-associative. Proposition 7.3. Let R be a commutative ring with unity of characteristic not equal to 2. Then R[Rn] is not power-associative for n> 3.

Proof. Let n> 3 and Rn = {a0, a1, . . . , an−1}. The product in Rn is deﬁned by the rule

ai.aj = a2j−i, where the indexes are taken by modulo n. Consider an element u = a0 + 2a1 ∈ R[Rn]. Then we have 2 u = a0 + 4a1 + 2a2 + 2an−1, and hence 2 u .u = 5a0 + 8a1 + 2a2 + 8an−3 + 4an−2 + 4an−1. On the other hand 2 u.u = a0 + 8a1 + 4a2 + 4a3 + 2a4 + 4an−3 + 2an−2 + 2an−1. 2 We see that if n > 3, then the coeﬃcient of a0 in u .u equals to 5, but the coeﬃcient of a0 in 2 2 2 u.u equals to 1. Hence, u .u 6= u.u , and the ring R[Rn] is not power-associative. We conclude with the following questions. Question 7.4. Let X and Y be two racks such that R[X] =∼ R[Y ]. Does it follow that X =∼ Y ?

Question 7.5. Let X and Y be two racks with XR(X) =∼ XR(Y ). Does it follow that X =∼ Y ?

Acknowledgement. The results given in sections 3, 6 and 7 are supported by the Russian Science Foundation grant 16-41-02006, and the results in 4 and 5 are supported by the DST grant INT/RUS/RSF/P-2 and SERB MATRICS grant. QUANDLE RINGS 19

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Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia. 20 VALERIY G. BARDAKOV, INDER BIR S. PASSI, AND MAHENDER SINGH

Novosibirsk State Agrarian University, Dobrolyubova street, 160, Novosibirsk, 630039, Russia. E-mail address: [email protected]

Center for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India.

Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India. E-mail address: [email protected]